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Metamath Proof Explorer

Theorem cxple3

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016)

Ref Expression
Assertion cxple3 ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 cxplt3 ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ) → ( 𝐶 < 𝐵 ↔ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )
2 1 ancom2s ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐶 < 𝐵 ↔ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )
3 2 notbid ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( ¬ 𝐶 < 𝐵 ↔ ¬ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )
4 lenlt ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵 ) )
5 4 adantl ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐵 ≤ 𝐶 ↔ ¬ 𝐶 < 𝐵 ) )
6 rpcxpcl ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ+ )
7 6 ad2ant2rl ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ+ )
8 rpcxpcl ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )
9 8 ad2ant2r ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )
10 rpre ⊢ ( ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ+ → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ )
11 rpre ⊢ ( ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ )
12 lenlt ⊢ ( ( ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ ∧ ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ ) → ( ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ↔ ¬ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )
13 10 11 12 syl2an ⊢ ( ( ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ+ ∧ ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ ) → ( ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ↔ ¬ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )
14 7 9 13 syl2anc ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ↔ ¬ ( 𝐴 ↑𝑐 𝐵 ) < ( 𝐴 ↑𝑐 𝐶 ) ) )
15 3 5 14 3bitr4d ⊢ ( ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 < 1 ) ∧ ( 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) ) → ( 𝐵 ≤ 𝐶 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐵 ) ) )